According to Shannon formula C=W·log2(1+SIN), where C is a channel capability, W is channel bandwidth, S is signal power and N is noise power, it can be seen that the channel capability is proportional to the bandwidth, therefore the most effective way to improve the channel capability is to increase the bandwidth. In addition, it can be seen that the channel capability can be also improved by increasing the signal power.
In a current communication system, different information is carried to different frequency bands by using carrier modulation technologies and transmitted on the frequency bands, the essence of which is to fully utilize bandwidth resources to improve the channel capability. FIG. 1 shows a current typical carrier modulation principle. The real part and the imaginary part of a baseband complex signal are respectively multiplied by carriers cos(ωt) and sin(ωt), accumulated and then transmitted. This process can be expressed by the following formula: sBP(t)=Re{sLP(t)eiωt}, where sBP(t) is a carrier modulated signal, eiωt is a complex carrier signal, sLP(t) is a baseband complex signal and Re represents to take the real part. The principle of this formula is that the multiplication of time domain signals is equal to convolution of frequency domain signals, and a baseband signal is shifted to a carrier frequency band via a convolution process of a carrier frequency signal and the baseband signal. Obviously, in the current carrier modulation method, although a baseband signal is presented by a complex number and a carrier signal is also represented by a complex number, what is transmitted finally is only the real part of a carrier modulated signal. Therefore, the real signal is transmitted, which is called real carrier modulation herein.
Actually, the current real carrier modulation method has multiplied the waste of frequency spectrum resources and the loss of signal energy, mainly because of the lack of a proper understanding and improper use of negative frequencies.
Firstly, negative frequencies do exist. As shown in FIG. 2, an angle of counterclockwise rotation is defined as +θ, and an angle of clockwise rotation is defined as −θ, then it can be learned based on the definition
  ω  =            ⅆ      θ              ⅆ      t      of an angular frequency that a negative angular frequency
      -    ω    =            ⅆ              (                  -          θ                )                    ⅆ      t      is generated by a “negative angle” instead of a “negative time”. Therefore, as a matter of fact, the positive and negative frequencies only represent that there are rotations in two different directions on a plane. In essence, the positive and negative rotations exist because the plane has two surfaces. The positive frequencies whose rotation directions accord with a right-hand rule are defined as right rotating frequencies herein which are called right frequencies for short. The negative frequencies whose rotation directions accord with a left-hand rule are defined as left rotating frequencies herein which are called left frequencies for short. Unless otherwise referred to, the positive and negative frequencies, the positive and negative frequency bands, and the positive and negative spectrums etc. in the existing technologies are replaced with terms such as left and right frequencies, left and right frequency bands, and left and right spectrums etc. hereinafter.
So far, whether in teaching materials or in engineering implementation, the defined available bandwidths (also known as work frequency bands) are within the range of right spectrums with positive signs, while left spectrums are abandoned selectively because of the negative signs in the mathematical expressions. As shown in FIG. 3 (protocol contents, the original text of which is English), frequency spectrum resources with negative signs are completely neglected even in the frequency band division of the most cutting-edge Long Term Evolution (LTE) communication protocol.
While understanding the natural existence of left frequencies, how to distinguish the left and right frequencies, or how to describe these two rotations on a plane? Euler's formula gives the answer: e±iωt=cos(ωt)±i sin(ωt). As shown in FIG. 4, e−iωt and eiωt represent a clockwise rotation curve and a counterclockwise rotation curve respectively, corresponding to the left and right frequency signals. Although the left and right frequency signals can be easily distinguished in a “time-complex number” space, the projections of the left and right frequency signals are all real signals cos(ωt) apparently on a “time-real part” plane, i.e. Re{e−iωt}=Re{e+iωt}=cos(ωt). Therefore, when a real signal appears, it cannot be distinguished whether it is the projection of a left frequency signal or the projection of a right frequency signal. Speaking from the probability, a left frequency signal and a right frequency signal is probabilistically-equivalent, i.e. both the probability of a left frequency signal and the probability of a right frequency signal are 1/2, i.e. cos(ωt)=(e−iωt+e+iωt)/2. Therefore, real signals with only one degree of freedom are incomplete. Unambiguous description of a frequency signal at least requires a complex signal with two degrees of freedom. In other words, a complete description of a frequency signal should be in a complex-number form. In the complete description, the left and right frequencies e−iωt and eiωt in the complex-number form are two completely independent distinguishable frequencies and can carry completely independent information.
As analyzed above, real signals generated by real carrier modulation actually cause ambiguity of left and right frequencies, thus the left and right frequency bands are both occupied, and information on the left and right frequency bands are in conjugate symmetry and not independent. FIG. 5 shows a spectrum shifting in real carrier modulation, wherein the abscissa represents a frequency ω, the ordinate represents amplitude F(ω) and ωC represents a carrier frequency. By the way, in the real carrier modulation mode, since two-dimensional complex signals are observed from incomplete one-dimensional real signals, a left frequency band caused by the real carrier modulation mode has brought great confusion to persons who do not know the meaning and function of the left frequency band, and may be erroneously assumed to be only a mirror image which does not really exist. A more serious point of view regards signals of the left frequency band harmful, thus bringing about many methods such as “mirror image inhibition”, and etc.
Currently, received signals are regarded as real signals during demodulation, therefore multiplication, i.e. frequency band shifting is performed for real signals only. Generally, a right frequency band is shifted to a baseband. In this way, a left frequency band is shifted to a position which is distant doubly from the baseband, and all information of the left frequency band is erased after being filtered by the baseband. Although the mirror image information of the left frequency band is redundant, the loss is multiplied actually when the mirror image information is abandoned directly. FIG. 6 shows a spectrum shifting in real carrier demodulation, wherein the abscissa represents a frequency ω, the ordinate represents amplitude F(ω) and ωC represents a carrier frequency. FIG. 7 shows an energy loss in a process from transmitting a signal to receiving the signal. A complete complex signal is a left-rotating or right-rotating plane signal (a); after the signal undergoes the grating effect (b) of real carrier modulation, and the projection effect (c) of a receiving antenna, the loss of the actually received signal energy may be quadrupled or more. Luckily, such incomplete real carrier demodulation is applied because the information carried in the left and right frequencies is conjugate mirror information, thus it is the same to receive the information on the left frequency even if a demodulation end is confused with the left and right frequencies. It only needs to exchange the I and Q data to mirror the information back, that's why many instruments are provided with an option for performing I, Q exchange for received signals.
It can be seen from the frequency band shifting process in the modulation and demodulation above that a frequency is actually a relative value which changes with the change of a reference frequency. The reference frequency here refers to a modulation and demodulation frequency and only the distance between the frequencies, i.e. the frequency band has an absolute meaning, which proves the actual existence of “negative frequencies” from another perspective.
To sum up, because of the natural bias to a left frequency, frequency spectrum resources of the left frequency are neglected by wireless, wire, optical fiber, radar and other bandwidth definitions in all current communication systems, thus half of the frequency spectrum resources are wasted. At the same time, the current real carrier modulation also make the left and right frequency bands occupied and the current real carrier demodulation also make the left frequency signal energy or the right frequency signal energy abandoned.